Recently, a classification of quantum states from more general non-signalling bipartite correlations has been obtained: quantum states correspond with those correlations, which satisfy (i) an extension of the no-signalling principle to dilations, and (ii) a relative consistency condition between the canonical unitary evolution in the respective subsystems (see Def. 1, Thm. 2 and Def. 2, Thm. 3 in [1]}). Correlations under (i) (but not necessarily (ii)) correspond with decomposable maps under the Choi-Jamiołkowski isomorphism in Eq. (REF ).We remark that for \(\mathrm {dim}(\mathcal {H}_\mathcal {A}) = 2\)  and \(\mathrm {dim}(\mathcal {H}_\mathcal {B}) = 2,3\)  every positive map \(\phi : \mathcal {A}\rightarrow \mathcal {B}\)  is decomposable [2]}, [3]}, which implies necessity of the PPT criterion in those dimensions [4]} (see also Thm. REF  below).
